>How far can I run a network cable from my wireless router to a PC before I would see a reasonable degradation of the internet signal strength?

The total length of twisted pair wire segments between a PC and a hub or between two PC's should not exceed 100 Meters (329 feet or about the length of a football field) for 100BASE-TX or 10BASE-T cable runs. The maximum length for 1000BaseTX twisted pair is presently 82 feet.

http://www.bluemax.com/techtips/networking/Wiring_Tips/Wiring100TX/letsmakesomecables.htm
http://www.dslreports.com/faq/5010
http://www.roboticsautomation.net/lan_std.htm

However, given those standards, keep in mind the things that may interfere or degradate the signal. For example, twisted pair carrying audio-visual signals should not be run within 1 foot of florescent lighting...

See more here:

http://www.avovercat5.com/faq.htm

I have learned over the years, regardless of what the experts say, keep it away from microwaves, ovens, lights, power sources, et al... (I spent numerous hours re-running cable in small insurance agencies and other small businesses as a last resort in troubleshooting only to find out that the "guaranteed certified" cable and where it ran "by the guaranteed certified installers" really was the problem)

I actually like this the best - it covers degradation and more:

http://www.geocities.com/SiliconValley/Peaks/7248/cabling.html

Sorry, forgot to mention (and you asked) degradation occurs supposedly at 300' but it can more or less depending on the environment.